A person pushes a 16.0-kg shopping cart at a constant velocity for a distance of 20.0 m. She pushes in a direction 28.0° below the horizontal. A 54.0-N frictional force opposes the motion of the cart.
To find the force applied by the person to move the shopping cart at a constant velocity, we need to consider the forces acting on the cart.
There are two forces acting horizontally:
1. Force applied by the person pushing the cart.
2. Frictional force opposing the motion.
Since the cart is moving at a constant velocity, the net force acting on it is zero. This means that the force applied by the person must be equal in magnitude and opposite in direction to the frictional force.
Using trigonometry, we can break down the force applied by the person into its horizontal and vertical components.
The horizontal component of the force is given by: F_horizontal = F_applied * cos θ
where F_applied is the force applied by the person and θ is the angle below the horizontal.
The vertical component of the force is given by: F_vertical = F_applied * sin θ
Given that the frictional force is 54.0 N and the angle θ is 28.0° below the horizontal, we can find the force applied by the person.
First, let's find the horizontal component of the force:
F_horizontal = F_applied * cos θ
F_horizontal = 54.0 N * cos(28.0°)
Next, let's find the vertical component of the force:
F_vertical = F_applied * sin θ
F_vertical = 54.0 N * sin(28.0°)
Since the cart is moving at constant velocity, the magnitude of the applied force must be equal to the magnitude of the frictional force. Therefore:
|F_applied| = |F_horizontal|
|F_applied| = sqrt(F_horizontal^2 + F_vertical^2)
Finally, let's calculate the force applied by the person:
|F_applied| = sqrt((54.0 N * cos(28.0°))^2 + (54.0 N * sin(28.0°))^2)
To solve this problem, we need to analyze the forces acting on the shopping cart.
Given information:
Mass of the shopping cart (m) = 16.0 kg
Distance traveled (d) = 20.0 m
Angle below the horizontal (θ) = 28.0°
Frictional force (Ff) = 54.0 N
Step 1: Determine the horizontal and vertical components of the force applied.
The horizontal component of the force applied (Fhx) can be calculated using the formula:
Fhx = Fh * cos(θ)
Where Fh is the total applied force (which we'll call Fa).
The vertical component of the force applied (Fvy) can be calculated using the formula:
Fvy = Fh * sin(θ)
Step 2: Calculate the total applied force.
Since the shopping cart is moving at a constant velocity, the total applied force (Fa) must balance out the opposing forces, which are the frictional force (Ff) and the vertical component of the force applied (Fvy).
Fa = Ff + Fvy
Step 3: Use the total applied force to calculate the horizontal component.
Using the equation from Step 1, we can substitute Fa to calculate Fhx:
Fhx = Fa * cos(θ)
Step 4: Calculate the work done.
The work done (W) can be calculated by multiplying the total applied force by the distance traveled:
W = Fa * d
Step 5: Substitute the values into the formulas to find the answers.
Let's calculate the horizontal and vertical components of the force applied.
Fhx = Fa * cos(θ)
Fhx = Fa * cos(28.0°)
Now, we can calculate the total applied force using the formula from Step 2.
Fa = Ff + Fvy
Fa = 54.0 N + Fvy
Substitute the values of Fhx and Fvy into the equation.
Fa = 54.0 N + Fhx * sin(θ)
Fa = 54.0 N + (Fa * cos(28.0°) * sin(28.0°))
Now, let's solve for Fa by moving the term with Fa to one side of the equation.
Fa - Fa * cos(28.0°) * sin(28.0°) = 54.0 N
Factor out Fa.
Fa(1 - cos(28.0°) * sin(28.0°)) = 54.0 N
Divide both sides by (1 - cos(28.0°) * sin(28.0°)).
Fa = 54.0 N / (1 - cos(28.0°) * sin(28.0°))
Finally, substitute the value of Fa into the equation for Fhx.
Fhx = Fa * cos(28.0°)
Now, we can calculate the work done using the formula from Step 4.
W = Fa * d
Substitute the values of Fa and d into the equation.
W = Fa * 20.0 m
Simplify further to find the final answer.